[Paper Review] Analysis and Application of Optimal Transport For Challenging Seismic Inverse Problems

Yunan Yang|arXiv (Cornell University)|Feb 4, 2019

Seismic Imaging and Inversion TechniquesEarth and Planetary Sciences55 references3 citations

TL;DR

This paper proposes using the quadratic Wasserstein distance (W₂) as a robust misfit measure in full-waveform inversion (FWI) to overcome three major challenges of traditional L²-based FWI: local minima, noise sensitivity, and poor recovery of subsurface velocity structures below deep reflectors. By leveraging optimal transport theory, W₂ captures both amplitude and phase differences, enabling stable, convex-like convergence and successful sub-reflection velocity recovery even with reflection-dominated data.

ABSTRACT

In seismic exploration, sources and measurements of seismic waves on the surface are used to determine model parameters representing geophysical properties of the earth. Full-waveform inversion (FWI) is a nonlinear seismic inverse technique that inverts the model parameters by minimizing the difference between the synthetic data from the forward wave propagation and the observed true data in PDE-constrained optimization. The traditional least-squares method of measuring this difference suffers from three main drawbacks including local minima trapping, sensitivity to noise, and difficulties in reconstruction below reflecting layers. Unlike the local amplitude comparison in the least-squares method, the quadratic Wasserstein distance from the optimal transport theory considers both the amplitude differences and the phase mismatches when measuring data misfit. We will briefly review our earlier development and analysis of optimal transport-based inversion and include improvements, for example, a stronger convexity proof. The main focus will be on the third "challenge" with new results on sub-reflection recovery.

Motivation & Objective

• Address the persistent challenge of local minima in L²-based full-waveform inversion (FWI), especially when low-frequency data are missing or the initial model is poor.
• Overcome the sensitivity of L² misfit to high-frequency noise, which amplifies artifacts and degrades inversion quality.
• Enable accurate recovery of low-wavenumber velocity structures beneath deep reflecting interfaces, where conventional FWI fails due to lack of diving waves and reliance on reflections.
• Demonstrate that W₂-based FWI achieves faster and more accurate model convergence than L²-FWI, particularly in reconstructing deeper subsurface features.
• Establish theoretical and numerical evidence that W₂ provides superior convexity and stability for seismic inverse problems compared to standard L² norms.

Proposed method

• Replace the standard L² misfit function in FWI with the quadratic Wasserstein distance (W₂), which measures data misfit by accounting for both amplitude and phase differences through optimal transport theory.
• Use a 1D explicit computation of W₂ via the cumulative distribution function (CDF) and its inverse, enabling efficient gradient computation in the optimization loop.
• Apply a sharper convexity theorem for W₂ under signal translations and dilations, proving improved convergence behavior for model updates.
• Implement a hierarchical error correction mechanism where low-frequency components of the data residual are corrected first, followed by high-frequency components, mimicking a multi-scale optimization strategy.
• Utilize the fact that W₂ becomes equivalent to a weighted H⁻¹ norm when the synthetic and observed data are close, ensuring stable and smooth convergence.
• Conduct large-scale numerical experiments on layered models and the 2004 BP salt model to validate W₂-FWI performance against L²-FWI in terms of data misfit, model error, and convergence speed.

Experimental results

Research questions

• RQ1 Can the quadratic Wasserstein distance (W₂) serve as a more convex and stable objective function than L² for full-waveform inversion in seismic exploration?
• RQ2 Does W₂-based FWI mitigate the issue of local minima in the absence of low-frequency data or with poor starting models?
• RQ3 How does W₂-based FWI perform in terms of robustness to noise compared to L²-based FWI?
• RQ4 Can W₂-based FWI successfully recover velocity structures below the deepest reflecting interface when only reflection data are available?
• RQ5 What is the hierarchical behavior of residual correction in W₂-FWI, and how does it relate to the recovery of low-wavenumber model components?

Key findings

• W₂-based FWI reduces model error by more than twice the amount achieved by L²-FWI in only 150 iterations, while L²-FWI shows only a 7% reduction in model error after 1800 iterations.
• The W₂ distance is equivalent to a weighted H⁻¹ norm when the synthetic and observed data are close, which ensures stable and smooth convergence during optimization.
• In the three-layer model test, W₂-FWI corrects low-frequency data residuals first and high-frequency residuals later, enabling systematic and stable model updates.
• The BP salt model inversion demonstrates that W₂-FWI successfully reconstructs the thickness and velocity of the layer below the deepest reflector, a task where L²-FWI fails due to cycle skipping and poor convergence.
• The Fourier analysis of residuals shows that W₂-FWI corrects smooth-mode errors early and oscillatory-mode errors only after the smooth components are minimized, indicating a hierarchical, multi-scale optimization behavior.
• The improved convexity theorem for W₂ under signal translation and dilation confirms its theoretical robustness and broader applicability to seismic data with varying time shifts and scaling.